The implementation of Logiweb

This paper describes the implementation of the ‘Logiweb’ system with emphasis on measures taken to support classical reasoning about programs. Logiweb is a system for authoring, storing, distributing, indexing, checking, and rendering of ‘Logiweb pages’. Logiweb pages may contain mathematical definitions, conjectures, lemmas, proofs, disproofs, theories, journal papers, computer programs, and proof checkers. Reading Logiweb pages merely requires access to the World Wide Web. Two example pages are available on http://yoa.dk/. Writing, checking, and publishing Logiweb pages requires Logiweb to be downloaded and installed. Logiweb comes with a hierarchy of features: Lemmas and proofs are stated in a theory named ‘Map Theory’, Map Theory is implemented on top of a calculus named ‘Logiweb sequent calculus’, and Logiweb sequent calculus is implemented on top of the ‘Logiweb reduction system’ (a version of λ-calculus). The Logiweb reduction system is implemented in the Logiweb core software which is currently implemented in Common Lisp. The levels above the Logiweb core software are defined on Logiweb pages, allowing users to use the features as they are or to define and publish new ones on new Logiweb pages. As an example, a user may want to use ZFC in place of Map Theory, in which case the easiest approach is to publish a Logiweb page that defines ZFC in Logiweb sequent calculus and proceed from there. The ‘base’ page on http://yoa.dk/, which is 180 pages long when printed out, was checked in 40 seconds. This is non-trivial to achieve for a proof checker implemented in lamdba calculus. The Logiweb sequent calculus is defined on the base page mentioned above. A user who wants to define e.g. ZFC set theory on top of that may publish a new page, call it ‘zfc’, and let the ‘zfc’ page reference the ‘base’ page. That makes all definitions on the ‘base’ page available to the ‘zfc’ page. After that, another user may state and prove lemmas about e.g. real numbers on a third page, call it ‘real’, which references the ‘zfc’ page. When the proofs on the ‘real’ page are checked, Logiweb will arrange that the ‘zfc’ and ‘base’ pages are available in a predigested form suitable for proof checking. Seen from the point of view of proof checking and publication, the World Wide Web has the drawback that once submitted pages can be modified after submission. In the example above, modification of the ‘base’ page could ruin the correctness of the ‘real’ page. To avoid problems with pages being modified, Logiweb implements its own referencing system which forces immutability upon once submitted pages. Once a Logiweb page is submitted, it cannot be changed, just like papers cannot change after publication.